Unified representation of the family of L-functions

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions $\mathbb{N}=\{1,2,\dots \}$, ${\mathbb{N}}_0=\{0,1,2,\dots \}=\mathbb{N}\cup \{0\}$, ${\mathbb{Z}}^{+}=\{1,2,3,\dots \}$, ${\mathbb{Z}}^{-}=\{-1,-2,\dots \}$. Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real number and ℂ denotes the set of complex numbers. We assume that $ln(z)$ denotes the principal branch of the multi-valued function $ln(z)$ with the imaginary part $\mathrm{\Im }(ln(z))$ constrained by $-\pi <\mathrm{\Im }(ln(z))\le \pi$.

where ($|t|<2\pi$ when $\beta =a$; $|t|<|blog(\frac{\beta }{a})|$ when $\beta \ne a$; $k\in {\mathbb{N}}_0$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\{0\}$).

For the special values of a, b, k, b and β, the polynomials ${\mathcal{Y}}_{n,\beta }(x;k,a,b)$ provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

(cf. [1–28]).

(cf. [1–28]).

(cf. [1–28]).

In terms of a Dirichlet character χ of conductor $f\in \mathbb{N}$, Ozden et al. [16] extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [[16], p.2783])

($|t|<2\pi$ when $\beta =a$; $|t|<|blog(\frac{\beta }{a})|$ when $\beta \ne a$; $k\in {\mathbb{N}}_0$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\{0\}$).

(cf. [1–26]).

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials ${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)$ in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate ${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)$ for negative integer values of n.

Throughout this section, we assume that $\beta \in \mathbb{C}$ with $|\beta |<1$ and $s\in \mathbb{C}$.

By using this functional equation, we arrive at the following theorem.

By using (7), we derive the following result.

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions $L_{\chi ,\beta }(s,x;k,a,b)$:

(11)

where ($\beta \in \mathbb{C}$ ($|\beta |<1$); $s\in \mathbb{C}$ ($\mathrm{\Re }(s)>1$)).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions $L_{\chi ,\beta }(s,x;k,a,b)$ as follows.

where ($\mathrm{\Re }(s)>1$, $\beta \in \mathbb{C}$ ($|\beta |<1$)).

(cf. [18]).

(cf. [27, 28]).

A relationship between the functions $L_{\chi ,\beta }(s,x;k,a,b)$ and ${\zeta }_{\beta }(s,x;k,a,b)$ is provided by the next theorem.

After some algebraic manipulations, we arrive at the desired result. □

(cf. [1–29]).

By using (15) and (14), we arrive at the following result.

By using Theorem 7 in [16], we get

By substituting (8) into the above, we arrive at the desired result. □

where $n\in {\mathbb{Z}}^{+}$ and $B_{n,\chi }$, the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that $L(s;\chi )$ is non-zero at $s=1$. Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at $s=1$ (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

From the above equation, we arrive at the following theorem.

where $q\in \mathbb{C}$ ($|q|<1$); ${\xi }^r=1$ ($r\in \mathbb{Z}$); $\xi \ne 1$. Observe that if $\xi =1$, then $L_{\xi ,q}^{(h)}(s,x;\chi )$ is reduced to the work of Kim [9].

Relationship between the function $L_{\chi ,\beta }(s,x;k,a,b)$ and $L_{\xi ,q}^{(h)}(s,x;\chi )$ is given as the following result.

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function $\mathrm{\Phi }(z,s,a)$, Hurwitz zeta function $\zeta (s,x)$ and L-function (cf. [1–30]); see also the references cited in each of these earlier works.